Bayesian inference for the proportion of true null hypotheses using minimum Hellinger distance

It is important that the proportion of true null hypotheses be estimated accurately in a multiple hypothesis context. Current estimation methods, however, are not suitable for high-dimensional data such as microarray data. First, they do not consider the (strong) dependence between hypotheses (or genes), thereby resulting in inaccurate estimation. Second, the unknown distribution of false null hypotheses cannot be estimated properly by these methods. Third, the estimation is affected strongly by outliers. In this paper, we find out the optimal procedure for estimating the proportion of true null hypotheses under a (strong) dependence based on the Dirichlet process prior. In addition, by using the minimum Hellinger distance, the estimation should be robust to any model misspecification as well as to any outliers while maintaining efficiency. The results are confirmed by a simulation study, and the newly developed methodology is illustrated by a real microarray data.     

倒向随机微分方程中非参数估计的核函数选择

比较了多种类型的核函数下倒向随机微分方程（BSDE）中生成元z的非参数估计方法,利用不同的核函数估计BSDE中的生成元z的非参数估计,在均方误差意义下比较了8种不同的核函数下得到的BSDE的生成元z的非参数估计的精度,统计分析结果显示Gaussian核函数下的估计效果最好。     

Estimation in second order branching processes with application to swine flu data

Abstract

This paper discusses inferential issues related to estimation of offspring mean and variance in a second order branching process, when both the offspring distributions are assumed to have identical mean and variance. Estimating equation approach is used to find the estimator of the offspring mean and the fact that a second order branching process model can be modeled as an autoregressive process is utilized to obtain the estimator of the offspring variance. Both the estimators are shown to be consistent and asymptotically normal. The second order branching process model is applied to H1N1 data for Pune, India, and Mexico and is found to be a suitable model. The estimates obtained from this model are used to compute the proportion of vaccination required for elimination of the disease.     

Large-sample comparisons of calibration procedures when both measurements are subject to error: the unreplicated case

A predictive functional relationship model is presented for the calibration problem in which the standard as well as the nonstandard measurements are subject to error. For the estimation of the relationship between the two measurements, the ordinary least squares and maximum likelihood estimation methods are considered, while for the prediction of unknown standard measurements we consider direct and inverse approaches. Relative performances of those calibration procedures are compared in terms of the asymptotic mean square error of prediction.     

On the estimation of missing values in AR(1) model with exponential innovations

We consider estimation of a missing value for a stationary autoregressive process of order one with exponential innovations and compare two methods of estimation of the missing value, with respect to Pitman's measure of closeness (PMC).     

Smoothing techniques via the bezier curve

Although the Bezier curve is very popular in the area of computational graphics it has rarely been used by statisticians. In this paper we develop methods and techniques for use of the Bezier curve in estimation of density and regression function. Also, asymptotic mean integrated square error for both estimators are derived. Comparisons with kernel estimator are conducted using simulation.     

Analyzing type II censored data obtained from repetitious experiments

Experimental design and Taguchi's parameter design are widely employed by industry to optimize the process/product. However, censored data are often observed in product lifetime testing during the experiments. After implementing a repetitious experiment with type II censored data, the censored data are usually estimated by establishing a complex statistical model. However, using the incomplete data to fit a model may not accurately estimates the censored data. Moreover, the model fitting process is complicated for a practitioner who has only limited statistical training. This study proposes a less complex approach to analyze censored data, using the least square estimation method and Torres's analysis of unreplicated factorials with possible abnormalities. This study also presents an effective method to analyze the censored data from Taguchi's parameter design using least square estimation method. Finally, examples are given to illustrate the effectiveness of the proposed methods.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Interval estimation for a binomial proportion: a bootstrap approach

This paper discusses the classic but still current problem of interval estimation of a binomial proportion. Bootstrap methods are presented for constructing such confidence intervals in a routine, automatic way. Three confidence intervals for a binomial proportion are compared and studied by means of a simulation study, namely: the Wald confidence interval, the Agresti–Coull interval and the bootstrap-t interval. A new confidence interval, the Agresti–Coull interval with bootstrap critical values, is also introduced and its good behaviour related to the average coverage probability is established by means of simulations.     

Robust Linearized Ridge M-estimator for Linear Regression Model

In the multiple linear regression, multicollinearity and outliers are commonly occurring problems. They produce undesirable effects on the ordinary least squares estimator. Many alternative parameter estimation methods are available in the literature which deals with these problems independently. In practice, it may happen that the multicollinearity and outliers occur simultaneously. In this article, we present a new estimator called as Linearized Ridge M-estimator which combats the problem of simultaneous occurrence of multicollinearity and outliers. A real data example and a simulation study is carried out to illustrate the performance of the proposed estimator.     

Numerical Algorithms Group Manual Mark 7

We consider the estimation of the 90 and 95 percentiles of a normal distribution and also the construction of one-sided 90% and 95% -normal ranges. Three methods are proposed -the sample percentile method, and two based on kernel estimates of the density function using Fryer's method and the leaving-one-out method for choosing a smoothing parameter.

A simulation study compares the methods in terms of bias, variance and mean square error of the population percentile estimates and of the eovers of the consequent normal ranges.     

Mean estimation through proportion estimation

In this work a device which changes the problem of mean estimation into that of proportion estimation is proposed. The device consists of perturbing the observations. The goal of the work is the construction of conservative confidence intervals for means. Three applications are given: (1) proportion estimation in the context of cluster random sampling, (2) differences of proportions of a multinomial population and (3) variance estimation.     

Modification of GEE1 and linear mixed-effects models for heteroscedastic longitudinal Gaussian data

A characterization of GLMs is given. Modification of the Gaussian GEE1, modified GEE1, was applied to heteroscedastic longitudinal data, to which linear mixed-effects models are usually applied. The modified GEE1 models scale multivariate data to homoscedastic data maintaining the correlation structure and apply usual GEE1 to homoscedastic data, which needs no-diagnostics for diagonal variances. Relationships among multivariate linear regression methods, ordinary/generalized LS, naïve/modified GEE1, and linear mixed-effects models were discussed. An application showed modified GEE1 gave most efficient parameter estimation. Correct specification of the main diagonals of heteroscedastic data variance appears to be more important for efficient mean parameter estimation.     

Parametric Estimator of Linear Model with Interval-Censored Data

In this article, we consider the estimation of regression parameters in linear model in the presence of interval-censored data. When the response variable is interval-censored, the traditional methods can not be used to estimate the parameters directly. In this article, unbiased transformation is carried out and a new random variable which has the same expectation as the function of the response variable is established. With the regression analysis for the constructed statistic we conclude the estimator by least square method.     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Estimation of a rare sensitive attribute in a stratified two-stage unrelated randomized response model by using Poisson probability distribution

The present article deals with the estimation of mean number of respondents who possess a rare sensitive character in presence of known and unknown proportion of a rare unrelated non-sensitive attribute by using the Poisson probability distribution in stratified random sampling as well as in stratified random double sampling. The variance of rare sensitive character is also derived under proportional and optimal allocation methods in stratified random sampling when stratum sizes are known and unknown. The properties of the suggested estimation procedures have been deeply examined. The proposed model is found to be dominant over Lee et al. [Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution. Statistics. 2013;47:575–589] model. Numerical illustrations are presented to support the theoretical results. Results are analysed and suitable recommendations are put forward to the survey practitioners.     

A nonparametric mixture approach to density and null proportion estimation in large-scale multiple comparison problems

A new method for estimating the proportion of null effects is proposed for solving large-scale multiple comparison problems. It utilises maximum likelihood estimation of nonparametric mixtures, which also provides a density estimate of the test statistics. It overcomes the problem of the usual nonparametric maximum likelihood estimator that cannot produce a positive probability at the location of null effects in the process of estimating nonparametrically a mixing distribution. The profile likelihood is further used to help produce a range of null proportion values, corresponding to which the density estimates are all consistent. With a proper choice of a threshold function on the profile likelihood ratio, the upper endpoint of this range can be shown to be a consistent estimator of the null proportion. Numerical studies show that the proposed method has an apparently convergent trend in all cases studied and performs favourably when compared with existing methods in the literature.     

A mover-stayer mixture of Markov chain models for the assessment of dedifferentiation and tumour progression in breast cancer

SUMMARY Malignancy grade is a histological measure of attributes related to a breast tumour's aggressive potential. It is not established whether the grade is an inate characteristic which remains unchanged throughout the tumour's development or whether it evolves as the tumour grows. It is likely that a proportion of tumours have the potential to evolve, and so a statistical method was required to assess this hypothesis and, if possible, to estimate the proportion with the potential for evolution. Therefore, a mover-stayer mixture of Markov chain models was developed, with the complication that 'movers' were unobservable because tumours were excised on diagnosis. A quasi-likelihood method was used for estimation. The methods are demonstrated using data from the Swedish twocounty trial of breast-cancer screening.     

Matching pseudocounts for interval estimation of binomial and Poisson parameters

ABSTRACT

For interval estimation of a binomial proportion and a Poisson mean, matching pseudocounts are derived, which give the one-sided Wald confidence intervals with second-order accuracy. The confidence intervals remove the bias of coverage probabilities given by the score confidence intervals. Partial poor behavior of the confidence intervals by the matching pseudocounts is corrected by hybrid methods using the score confidence interval depending on sample values.     

Improved estimation methods for unrelated question randomized response techniques

The randomized response technique (RRT) is an important tool, commonly used to avoid biased answers in survey on sensitive issues by preserving the respondents’ privacy. In this paper, we introduce a data collection method for survey on sensitive issues combining both the unrelated-question RRT and the direct question design. The direct questioning method is utilized to obtain responses to a non sensitive question that is related to the innocuous question from the unrelated-question RRT. These responses serve as additional information that can be used to improve the estimation of the prevalence of the sensitive behavior. Furthermore, we propose two new methods for the estimation of the proportion of respondents possessing the sensitive attribute under a missing data setup. More specifically, we develop the weighted estimator and the weighted conditional likelihood estimator. The performances of our estimators are studied numerically and compared with that of an existing one. Both proposed estimators are more efficient than the Greenberg's estimator. We illustrate our methods using real data from a survey study on illegal use of cable TV service in Taiwan.